A higher order discontinuous Galerkin, global shallow water model: Global ocean tides and aquaplanet benchmarks

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The development of models of the ocean tides with higher resolution near coastlines and coarser resolution offshore, has been required to account for the significant impacts of coastline configuration and bathymetry (associated with sea level rise) on both the amplitude and phase of tidal constituents. This capability becomes especially important in the context of tidal analyses at times in the past when sea levels are known to have differed significantly from present. Here we present a novel global model based on the discontinuous Galerkin disretization of the shallow water equations that employs third order Runge–Kutta time stepping on unstructured triangular grids. The model has been efficiently parallelized and is thereby shown to achieve essentially perfect linear scaling which makes it suitable for the generation of extremely high resolution results in local regions of interest.The paper includes a detailed development of the mathematical and numerical framework which is first tested in the context of analyses of a series of well established aquaplanet benchmarks for the shallow water equations on the sphere. These benchmarks include: (1) steady state nonlinear geostrophic flow in the context of an hp convergence study, (2) Rossby wave response arising from a geostrophic flow impinging on localized topography and (3) development of barotropic instability in a perturbed balanced zonal flow. For our target tidal applications the basic shallow water system is extended to include the influence of internal tide-related drag in the deep ocean as well as the drag in shallow marginal seas together with the influence of gravitational self-attraction and loading. Global tidal simulations with various offshore and coastal resolutions are compared with the standard benchmark based upon satellite altimetry data. Initial investigations of the M2 tidal amplitude, phase and energy budget are provided and shown to be highly satisfactory at the level of resolution for which results are provided.